1. Warm Up An experiment consists of rolling a fair number cube with faces numbered 2, 4, 6, 8, 10, and 12. Find each probability. 1. P(rolling an even number) 2. P(rolling a prime number) 3. P(rolling a number > 7) 1 1 6 1 2 Course 3 10-8 Counting Principles

2. Learn to find the number of possible outcomes in an experiment. Course 3 10-8 Counting Principles

3. Vocabulary Fundamental Counting Principle tree diagram Addition Counting Principle Insert Lesson Title Here Course 3 10-8 Counting Principles

4. Course 3 10-8 Counting Principles

5. License plates are being produced that have a single letter followed by three digits. All license plates are equally likely. Additional Example 1A: Using the Fundamental Counting Principle Find the number of possible license plates. Use the Fundamental Counting Principal. letterfirst digit second digit third digit 26 choices10 choices 26 10 10 10 = 26,000 The number of possible 1-letter, 3-digit license plates is 26,000. Course 3 10-8 Counting Principles

6. Additional Example 1B: Using the Fundamental Counting Principal Find the probability that a license plate has the letter Q. 1 10 10 10 26,000 = 1 26  0.038 P(Q ) = Course 3 10-8 Counting Principles

7. Additional Example 1C: Using the Fundamental Counting Principle Find the probability that a license plate does not contain a 3. First use the Fundamental Counting Principle to find the number of license plates that do not contain a 3. 26 9 9 9 = 18,954 possible license plates without a 3 There are 9 choices for any digit except 3. P(no 3) = = 0.729 26,000 18,954 Course 3 10-8 Counting Principles

8. Social Security numbers contain 9 digits. All social security numbers are equally likely. Check It Out: Example 1A Find the number of possible Social Security numbers. Use the Fundamental Counting Principle. Digit123456789 Choices10 10 10 10 10 10 10 10 10 10 = 1,000,000,000 The number of Social Security numbers is 1,000,000,000. Course 3 10-8 Counting Principles

9. Check It Out: Example 1B Find the probability that the Social Security number contains a 7. P(7 _ _ _ _ _ _ _ _) = 1 10 10 10 10 10 10 10 10 1,000,000,000 = = 0.1 10 1 Course 3 10-8 Counting Principles

10. Check It Out: Example 1C Find the probability that a Social Security number does not contain a 7. First use the Fundamental Counting Principle to find the number of Social Security numbers that do not contain a 7. P(no 7 _ _ _ _ _ _ _ _) = 9 9 9 9 9 9 9 9 9 1,000,000,000 P(no 7) = ≈ 0.4 1,000,000,000 387,420,489 Course 3 10-8 Counting Principles

11. The Fundamental Counting Principle tells you only the number of outcomes in some experiments, not what the outcomes are. A tree diagram is a way to show all of the possible outcomes. Course 3 10-8 Counting Principles

12. Additional Example 2: Using a Tree Diagram You have a photo that you want to mat and frame. You can choose from a blue, purple, red, or green mat and a metal or wood frame. Describe all of the ways you could frame this photo with one mat and one frame. You can find all of the possible outcomes by making a tree diagram. There should be 4 2 = 8 different ways to frame the photo. Course 3 10-8 Counting Principles

13. Additional Example 2 Continued Each “branch” of the tree diagram represents a different way to frame the photo. The ways shown in the branches could be written as (blue, metal), (blue, wood), (purple, metal), (purple, wood), (red, metal), (red, wood), (green, metal), and (green, wood). Course 3 10-8 Counting Principles

14. Check It Out: Example 2 A baker can make yellow or white cakes with a choice of chocolate, strawberry, or vanilla icing. Describe all of the possible combinations of cakes. You can find all of the possible outcomes by making a tree diagram. There should be 2 3 = 6 different cakes available. Course 3 10-8 Counting Principles

15. Check It Out: Example 2 Continued The different cake possibilities are (yellow, chocolate), (yellow, strawberry), (yellow, vanilla), (white, chocolate), (white, strawberry), and (white, vanilla). white cake yellow cake chocolate icing vanilla icing strawberry icing chocolate icing vanilla icing strawberry icing Course 3 10-8 Counting Principles

16. Additional Example 3: Using the Addition Counting Principle The table shows the items available at a farm stand. How many items can you choose from the farm stand? None of the lists contains identical items, so use the Addition Counting Principle. Total Choices Course 3 10-8 Counting Principles ApplesPearsSquash+=+ ApplesPearsSquash MacintoshBoscAcorn Red DeliciousYellow BartlettHubbard Gold DeliciousRed Bartlett

17. Additional Example 3 Continued Course 3 10-8 Counting Principles T332+=+= 8 There are 8 items to choose from.

18. Check It Out: Example 3 The table shows the items available at a clothing store. How many items can you choose from the clothing store? None of the lists contains identical items, so use the Addition Counting Principle. Course 3 10-8 Counting Principles T-ShirtsSweatersPants Long SleeveWoolDenim Shirt SleeveCottonKhaki PocketPolyester Cashmere

19. Additional Example 3 Continued Course 3 10-8 Counting Principles T342+=+= 9 There are 9 items to choose from. Total ChoicesT-shirtsSweatersPants+=+

20. Lesson Quiz: Part I Personal identification numbers (PINs) contain 2 letters followed by 4 digits. Assume that all codes are equally likely. 1. Find the number of possible PINs. 2. Find the probability that a PIN does not contain a 6. 0.6561 6,760,000 Insert Lesson Title Here Course 3 10-8 Counting Principles

21. Lesson Quiz: Part II A lunch menu consists of 3 types of sandwiches, 2 types of soup, and 3 types of fruit. 3. What is the total number of lunch items on the t menu? 4. A student wants to order one sandwich, one t bowl of soup, and one piece of fruit. How many t different lunches are possible? 18 8 Insert Lesson Title Here Course 3 10-8 Counting Principles